Infinite-dimensional Compact Quantum Semigroup

TL;DR

This paper constructs a compact quantum semigroup structure on the Toeplitz algebra, explores its dual algebra's substructure related to measures on a circle, and investigates Haar functionals and connections to weak Hopf algebras.

Contribution

It introduces a novel compact quantum semigroup structure on the Toeplitz algebra and analyzes its dual, revealing new subalgebra structures and connections to weak Hopf algebra theory.

Findings

Existence of a compact quantum semigroup structure on the Toeplitz algebra.

Identification of a subalgebra isomorphic to measures on a circle with convolution.

Existence of Haar functionals in the dual algebra and subalgebra.

Abstract

In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra $\mathcal{T}^*$ is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between $\mathcal{T}$ and the structure of weak Hopf algebra.